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Journal of mathematics and computer science
14 (2015), 162-170
Modification of the HPM by using optimal Newton interpolation polynomial
for quadratic Riccati differential equation
F. Ghomanjania, F. Divandarb
a; Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
b; Department of Control, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
[email protected]
Article history:
Received November 2014
Accepted December 2014
Available online December 2014
Abstract
In this work, an efficient modification of the homotopy analysis method by using optimal Newton
interpolation polynomials is given for the approximate solutions of the Riccati differential equations. This
presented method can be applied to linear and nonlinear models. Examples show that the method is
effective.
Keywords: quadratic Riccati differential equation, modification of the HPM, Newton interpolation.
1. Introduction
Riccati differential equations are a class of nonlinear differential equations of much importance, and play
a significant role in many fields of applied science [1]. The Riccati differential equation is named after the
Italian nobleman Count Jacopo Francesco Riccati (1676-1754). The applications of this equation may be
found not only in random processes, optimal control, and diffusion problems [2] but also in stochastic
realization theory, optimal control, robust stabilization, network synthesis and financial mathematics.
Solitary wave solutions of a nonlinear partial differential equation can be expressed as a polynomial in
two elementary functions satisfying a projective Riccati equation [3]. Therefore, one has to go for
numerical techniques or approximate approaches for getting its solution. Recently various iterative
methods are employed for the numerical and analytical solution of functional equations such as
Adomian's decomposition method (ADM) [4, 5], homotopy analysis method (HAM) [6], homotopy
perturbation method (HPM) [7], variational iteration method (VIM) [8], and differential transform method
(DTM) [9]. In [10], Liao has shown that HPM equations are equivalent to HAM equations when ћ= -1,
and too, matrix differential transform method for solving of Riccati types matrix differential equations
[11]. In this work, we introduce a new modification the HPM using optimal Newton interpolation
polynomials. The schemes are tested for some examples.
This study is organized as follows: In section 2, we present the standard HAM. In section 3, we present
the modification technique of HAM. In section 4, the method is applied to a variety of examples to show
efficiency and simplicity of the method.
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
2. Basic ideas of HAM
For the convenience of the reader, we will first present a brief account of HAM [12]. Let us consider the
following differential equation
[ ( )]
(1)
where N is a nonlinear operator, u(x, t) is an unknown function, and x and t denote spatial and temporal
independent variables. By means of homotopy analysis method, we first construct the so-called zerothorder deformation equation
[ ( )]
(
) [ (
)
( )]
(2)
where p ϵ [0; 1] is the embedding parameter, ћ is a non-zero auxiliary parameter, L is an auxiliary linear
operator, u0(x; t) is an initial guess of u(x, t) and φ(x, t; p) is a unknown function. Obviously, when p = 0
and p = 1, it holds
(
)
( )
(
)
( )
(3)
Thus, as p increases from 0 to 1, the solution φ(x, t; p) varies from the initial guess u0(x; t) to the exact
solution u(x, t). Expanding φ(x, t; p) in Taylor series with respect to p, we have
(
)
(
)
Where
(
(
∑
(
)
)
)
(4)
|
(5)
If the auxiliary linear operator, the initial guess, the auxiliary parameter ћ, and the auxiliary function are
properly chosen, series 4 converges at p = 1, then we have
(
)
∑
(
)
(6)
which must be one of solutions of original nonlinear equations. According to definition (6), the governing
equation can be deduced from the zero-order deformation (4). Define the vector
⃗ (
)
{
(
)
(
)
(
)
(
)}
(7)
Differentiating (4) m-times with respect to the embedding parameter p and then setting p = 0 and finally
dividing them by m!, we have the so-called mth-order deformation equation
[
(
)
(
where
(
)
)]
(⃗
(
(
)
)
)
(8)
|
(9)
163
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
{
10
3. Modified homotopy analysis method
For MHAM consider the following general nonlinear differential equation;
[ (
)]
( )
( )
(11)
where A is a general differential operator, f(x) is a known analytic function. In this study, alternatively,
f(x) is approximated by optimal Newton interpolation, Pf(x) over the interval [a, b],
( )
( )
∑ [
( )
] ∏(
)
where xks are the roots of the (ϑ + 1)st Chebyshev polynomial of the first kind Tϑ+1(x) in [-1, 1] translated
onto the interval [a, b] given by
(
)
And
[
]
[ ]
[
]
[
]
( )
4. Numerical Examples
In this section, we demonstrate the effectiveness of the proposed modification of the HAM by applying it
to two nonlinear and one linear problems. For each example, the M-term approximation of u(x) and the
exact solution is presented. Moreover, all numerical results obtained by the modification of the HAM
using optimal Newton interpolation polynomial, uM, are compared with the exact solutions. The algorithm
are performed by Matlab version 7.6.0.324 (R2008a) with 10 digits precision.
Example 1. Consider the following quadratic Riccati differential equation taken from [1]
( )
( )
( )
subject to the initial condition u(0) = 1 and the exact solution u(x) = et. To solve (12) by HAM, we
construct the following homotopy:
(
)(
)
(
( )
( )
( ))
(12)
(13)
Where
( )
(14)
Assume the solution of (12) to be in form:
164
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
,
(15)
Substituting (15) into (13) and collecting terms of the same power of p gives
( ))
(
(
)
(
)
(
)
(16)
The given initial value admits the use of
( )
To solve the above equations, we use the optimal Newton interpolation polynomial of f. For this reason,
by setting ϑ = 10, the optimal Newton interpolation polynomial of f becomes,
( )
( )
( )
∑ [
(
)
] ∏(
)
where
So, we have
( )
(17)
By substituting (17) into (16), we obtain ui(x), for i≥1, as
( )
165
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
( )
…
Here, for a given arbitrary natural number M,
∑
( )
It denotes the M-term approximation of the exact solution u(x) which is obtained using optimal Newton
interpolation of f. In Table 1, we give the numerical results of the exact solutions and the approximate
solutions obtained by the present method with ћ=-0.5, M = 10 and ϑ=10 in the interval 0 ≤ x ≤ 1. The
absolute errors for this solution are shown in Table 2. The graphs of approximated trajectory and exact
trajectory are plotted in Figure 1.
Table 1: Numerical results of the modified HAM with ћ=-0.5 for
Example 1
xi
exact solution u(xi)
Modified HAM M = 10
0.2
1.2214027581
1.2213091060
0.4
1.4918246976
1.4926294519
0.6
1.8221188003
1.8263387073
0.8
2.2255409284
2.2300568140
Table 2: Absolute errors of the modified HAM with ћ=-0.5 for Example 1
xi
M = 10
0.2
9.36521e-005
0.4
8.047543e-004
0.6
4.2199070e-003
0.8
4.5158856e-003
Example 2. Consider the following quadratic Riccati differential equation taken from [13]
( )
( )
( )
(18)
subject to the initial condition u(0) = 1 and the exact solution u(x) =1/(1+x). To solve (18) by HAM, we
construct a homotopy:
(
)(
)
(
( )
166
( )
( ))
(19)
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
Where
( )
(20)
Assume the solution of (18) to be in form:
,
(21)
using the optimal Newton interpolation polynomial of f, we have
( )
(22)
By substituting (22) into (19), we obtain ui(x), for i≥1, as
( )
( )
…
Here, for a given arbitrary natural number M,
∑
( )
It denotes the M-term approximation of the exact solution u(x) which is obtained using optimal Newton
interpolation of f.
In Table 3, we give the numerical results of the exact solutions and the approximate solutions obtained by
the present method with ћ = -1, M = 10 and ϑ = 10 in the interval 0 ≤ x ≤ 1. The absolute errors for this
solution are shown in Table 4. The graphs of approximated trajectory and exact trajectory are plotted in
Figure 2.
Table 3: Numerical results of the modified HAM with ћ=-1 for Example
2
xi
exact solution u(xi)
Modified HAM M = 10
0.2
0. 8333333333
0. 8333333345
0.4
0. 7142857142
0. 7142877171
0.6
0. 6250000000
0. 6251263624
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F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
0.8
0. 5555555555
0. 5578072670
Table 4: Absolute errors of the modified HAM with ћ=-1 for Example 2
xi
M = 10
0.2
1. 2e-009
0.4
2. 0029e-006
0.6
1. 263624e-004
0.8
2. 2517115e-003
Figure 1: comparison exact trajectory
approximated trajectory with presented method
for Example 1, with ћ=-0.5 and M = 10.
Figure 2: comparison exact trajectory
approximated trajectory with presented method
for Example 2, with ћ=-1 and M = 10.
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F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
Figure 3: The error graph of the modified HAM
with ћ=-0.5 and M = 10 for Example 1.
Figure 4: The error graph of the modified HAM
with ћ=-1 and M = 10 for Example 2.
169
F. Ghomanjani, F. Divandar/ J. Math. Computer Sci. 14 (2015), 162-170
5. conclusion
A simple and effective algorithm based on optimal Newton interpolation polynomial is presented which is
stated for solving quadratic Riccati differential equation. The method is computationally attractive, and
some examples are solved.
References
[1] S. Abbasbandy, A new application of He's variational iteration method for quadratic Riccati di_erential equation by using
adomian's polynomials, Comput. Appl. Math. 207 (2007) 59-63.
[2] W.T. Reid, Riccati differential equations, Mathematics in science and engineering, Academic Press, New York, 86.
[3] S. Abbasbandy, Iterated He's homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput.
175 (2006) 581-589.
[4] J. Biazar, H. Aminikhah, Study of convergence of homotopy perturbation method for systems of partial differential,
Computers Mathematics with Applications, 58 (2009) 2221-2230.
[5] J. Biazar, H. Ghazvini, Exact solutions for non-linear Schrodinger equations by He's homotopy perturbation method, Physics
Letters A, 366 (2007) 79-84.
[6] SJ. Liao, Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Champan and Hall, Boca Raton,
2003.
[7] H. Aminikhah, M. Hemmatnezhad, An effcient method for quadratic Riccati differential equation, Commun. Nonlinear Sci.
Numer. Simul. 15 (2010) 835-839.
[8] A. Ghorbani, S. Momani, An effective variational iteration algorithm for solving Riccati differential equations, Appl. Math.
Lett. 23 (2010) 922-927.
[9] H. Fatoorehchi, H. Abolghasemi, Improving the differential transform method: A novel technique to obtain the differential
transforms of nonlinearities by the Adomian polynomials, Applied Mathematical Modeling. 37 (2013) 6008-6017.
[10] S. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and
Computation, 169 (2005) 1186-1194.
[11] R. Abazari, Solution of Riccati types matrix differential equations using matrix differential transform method, Appl. Math.
Informatics. 27, 5-6 (2009) 1133-1143.
[12] N. Alam khan, M. Jamil, and A. Ara, Approximate Solutions to Time-Fractional Schrodinger Equation via Homotopy
Analysis Method, ISRN Mathematical Physics, 197068 (2012) 1-11.
[13]M. Lakestani, M. Dehghan, A new application of He's variational iteration method for quadratic Riccati differential equation
by using Adomian's polynomials, Computer Physics Communications, 181 (2010) 957-966.
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